3.20 \(\int x^5 (2+3 x^2) (5+x^4)^{3/2} \, dx\)

Optimal. Leaf size=83 \[ \frac{3}{14} \left (x^4+5\right )^{5/2} x^4-\frac{5}{24} \left (x^4+5\right )^{3/2} x^2-\frac{25}{16} \sqrt{x^4+5} x^2-\frac{1}{42} \left (18-7 x^2\right ) \left (x^4+5\right )^{5/2}-\frac{125}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

[Out]

(-25*x^2*Sqrt[5 + x^4])/16 - (5*x^2*(5 + x^4)^(3/2))/24 + (3*x^4*(5 + x^4)^(5/2))/14 - ((18 - 7*x^2)*(5 + x^4)
^(5/2))/42 - (125*ArcSinh[x^2/Sqrt[5]])/16

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Rubi [A]  time = 0.0588598, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1252, 833, 780, 195, 215} \[ \frac{3}{14} \left (x^4+5\right )^{5/2} x^4-\frac{5}{24} \left (x^4+5\right )^{3/2} x^2-\frac{25}{16} \sqrt{x^4+5} x^2-\frac{1}{42} \left (18-7 x^2\right ) \left (x^4+5\right )^{5/2}-\frac{125}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5*(2 + 3*x^2)*(5 + x^4)^(3/2),x]

[Out]

(-25*x^2*Sqrt[5 + x^4])/16 - (5*x^2*(5 + x^4)^(3/2))/24 + (3*x^4*(5 + x^4)^(5/2))/14 - ((18 - 7*x^2)*(5 + x^4)
^(5/2))/42 - (125*ArcSinh[x^2/Sqrt[5]])/16

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int x^5 \left (2+3 x^2\right ) \left (5+x^4\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (2+3 x) \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{3}{14} x^4 \left (5+x^4\right )^{5/2}+\frac{1}{14} \operatorname{Subst}\left (\int x (-30+14 x) \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{3}{14} x^4 \left (5+x^4\right )^{5/2}-\frac{1}{42} \left (18-7 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{5}{6} \operatorname{Subst}\left (\int \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac{5}{24} x^2 \left (5+x^4\right )^{3/2}+\frac{3}{14} x^4 \left (5+x^4\right )^{5/2}-\frac{1}{42} \left (18-7 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{25}{8} \operatorname{Subst}\left (\int \sqrt{5+x^2} \, dx,x,x^2\right )\\ &=-\frac{25}{16} x^2 \sqrt{5+x^4}-\frac{5}{24} x^2 \left (5+x^4\right )^{3/2}+\frac{3}{14} x^4 \left (5+x^4\right )^{5/2}-\frac{1}{42} \left (18-7 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{125}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=-\frac{25}{16} x^2 \sqrt{5+x^4}-\frac{5}{24} x^2 \left (5+x^4\right )^{3/2}+\frac{3}{14} x^4 \left (5+x^4\right )^{5/2}-\frac{1}{42} \left (18-7 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{125}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0543012, size = 72, normalized size = 0.87 \[ \frac{1}{6} x^2 \left (x^4+5\right )^{5/2}+\frac{3}{14} \left (x^4-2\right ) \left (x^4+5\right )^{5/2}-\frac{5}{48} \left (\sqrt{x^4+5} \left (2 x^4+25\right ) x^2+75 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(2 + 3*x^2)*(5 + x^4)^(3/2),x]

[Out]

(x^2*(5 + x^4)^(5/2))/6 + (3*(-2 + x^4)*(5 + x^4)^(5/2))/14 - (5*(x^2*Sqrt[5 + x^4]*(25 + 2*x^4) + 75*ArcSinh[
x^2/Sqrt[5]]))/48

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Maple [A]  time = 0.024, size = 73, normalized size = 0.9 \begin{align*}{\frac{ \left ( 3\,{x}^{4}-6 \right ) \left ({x}^{8}+10\,{x}^{4}+25 \right ) }{14}\sqrt{{x}^{4}+5}}+{\frac{{x}^{10}}{6}\sqrt{{x}^{4}+5}}+{\frac{35\,{x}^{6}}{24}\sqrt{{x}^{4}+5}}+{\frac{25\,{x}^{2}}{16}\sqrt{{x}^{4}+5}}-{\frac{125}{16}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(3*x^2+2)*(x^4+5)^(3/2),x)

[Out]

3/14*(x^4+5)^(1/2)*(x^4-2)*(x^8+10*x^4+25)+1/6*x^10*(x^4+5)^(1/2)+35/24*x^6*(x^4+5)^(1/2)+25/16*x^2*(x^4+5)^(1
/2)-125/16*arcsinh(1/5*x^2*5^(1/2))

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Maxima [A]  time = 1.42308, size = 171, normalized size = 2.06 \begin{align*} \frac{3}{14} \,{\left (x^{4} + 5\right )}^{\frac{7}{2}} - \frac{3}{2} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} - \frac{125 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{8 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}} - \frac{3 \,{\left (x^{4} + 5\right )}^{\frac{5}{2}}}{x^{10}}\right )}}{48 \,{\left (\frac{3 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{3 \,{\left (x^{4} + 5\right )}^{2}}{x^{8}} + \frac{{\left (x^{4} + 5\right )}^{3}}{x^{12}} - 1\right )}} - \frac{125}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{125}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(3*x^2+2)*(x^4+5)^(3/2),x, algorithm="maxima")

[Out]

3/14*(x^4 + 5)^(7/2) - 3/2*(x^4 + 5)^(5/2) - 125/48*(3*sqrt(x^4 + 5)/x^2 - 8*(x^4 + 5)^(3/2)/x^6 - 3*(x^4 + 5)
^(5/2)/x^10)/(3*(x^4 + 5)/x^4 - 3*(x^4 + 5)^2/x^8 + (x^4 + 5)^3/x^12 - 1) - 125/32*log(sqrt(x^4 + 5)/x^2 + 1)
+ 125/32*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 1.53446, size = 166, normalized size = 2. \begin{align*} \frac{1}{336} \,{\left (72 \, x^{12} + 56 \, x^{10} + 576 \, x^{8} + 490 \, x^{6} + 360 \, x^{4} + 525 \, x^{2} - 3600\right )} \sqrt{x^{4} + 5} + \frac{125}{16} \, \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(3*x^2+2)*(x^4+5)^(3/2),x, algorithm="fricas")

[Out]

1/336*(72*x^12 + 56*x^10 + 576*x^8 + 490*x^6 + 360*x^4 + 525*x^2 - 3600)*sqrt(x^4 + 5) + 125/16*log(-x^2 + sqr
t(x^4 + 5))

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Sympy [A]  time = 13.5729, size = 131, normalized size = 1.58 \begin{align*} \frac{x^{14}}{6 \sqrt{x^{4} + 5}} + \frac{3 x^{12} \sqrt{x^{4} + 5}}{14} + \frac{55 x^{10}}{24 \sqrt{x^{4} + 5}} + \frac{12 x^{8} \sqrt{x^{4} + 5}}{7} + \frac{425 x^{6}}{48 \sqrt{x^{4} + 5}} + \frac{15 x^{4} \sqrt{x^{4} + 5}}{14} + \frac{125 x^{2}}{16 \sqrt{x^{4} + 5}} - \frac{75 \sqrt{x^{4} + 5}}{7} - \frac{125 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(3*x**2+2)*(x**4+5)**(3/2),x)

[Out]

x**14/(6*sqrt(x**4 + 5)) + 3*x**12*sqrt(x**4 + 5)/14 + 55*x**10/(24*sqrt(x**4 + 5)) + 12*x**8*sqrt(x**4 + 5)/7
+ 425*x**6/(48*sqrt(x**4 + 5)) + 15*x**4*sqrt(x**4 + 5)/14 + 125*x**2/(16*sqrt(x**4 + 5)) - 75*sqrt(x**4 + 5)
/7 - 125*asinh(sqrt(5)*x**2/5)/16

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Giac [A]  time = 1.14673, size = 88, normalized size = 1.06 \begin{align*} \frac{1}{336} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (9 \, x^{2} + 7\right )} x^{2} + 72\right )} x^{2} + 245\right )} x^{2} + 180\right )} x^{2} + 525\right )} x^{2} - 3600\right )} + \frac{125}{16} \, \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(3*x^2+2)*(x^4+5)^(3/2),x, algorithm="giac")

[Out]

1/336*sqrt(x^4 + 5)*((2*((4*((9*x^2 + 7)*x^2 + 72)*x^2 + 245)*x^2 + 180)*x^2 + 525)*x^2 - 3600) + 125/16*log(-
x^2 + sqrt(x^4 + 5))